On the Critical Density for Percolation in Random Geometric Graphs

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density λ c for d-dimensional Poisson random geometric graphs in continuum percolation theory. In two-dimensional space with the Euclidean norm, simulation studies show λ c ≈ 1.44, while the best theoretical bounds obtained thus far are 0.696 < λ c < 3.372. By using a probabilistic analysis which incorporates clustering effects in random geometric graphs, we develop a new class of lower bounds for the critical density λ c for d-dimensional Poisson random geometric graphs. The lower bounds are the tightest known to date. In particular, for the two-dimensional case, the lower bound is substantially improved to λ c ≥ 0.833. For the three-dimensional case, we obtain λ c ≥ 0.45.